# A Hybrid Parallel Numerical Algorithm for Three-Dimensional Phase Field Modeling of Crystal Growth

## Main Article Content

## Abstract

We introduce a hybrid parallel numerical algorithm for solving the phase field formulation of the anisotropic crystal growth during solidification. The implementation is based on the MPI and OpenMP standards. The algorithm has undergone a number of efficiency measurements and parallel profiling scenarios. We compare the results for several variants of the algorithm and decide on the most efficient solution.

## Article Details

How to Cite

STRACHOTA, Pavel; BENEŠ, Michal.
A Hybrid Parallel Numerical Algorithm for Three-Dimensional Phase Field Modeling of Crystal Growth.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 23-32, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/391>. Date accessed: 20 sep. 2017.
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## References

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[16] M. PunKay. Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In NNIN REU Research Accomplishments, pages 116–117. University of Michigan, 2005.

[17] S. L. R. E. Napolitano. Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system. Phys. Rev. B, 70(21):214103, 2004.

[18] W. E. Schiesser. The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego, 1991.

[19] A. Schmidt. Computation of three dimensional dendrites with finite elements. J. Comput. Phys., 125:293–3112, 1996.

[20] P. Stenström, T. Joe, and A. Gupta. Comparative performance evaluation of cache-coherent NUMA and COMA architectures. In ISCA ’92 Proceedings of the 19th annual international symposium on Computer architecture, volume 20, pages 80–91. ACM New York, NY, USA, May 1992.

[21] P. Strachota. Analysis and Application of Numerical Methods for Solving Nonlinear ReactionDiffusion Equations. PhD thesis, Czech Technical University in Prague, 2012.

[22] P. Strachota and M. Beneš. A multipoint flux approximation finite volume scheme for solving anisotropic reaction-diffusion systems in 3D. In J. Fořt, J. Fürst, J. Halama, R. Herbin, and F. Hubert, editors, Finite Volumes for Complex Applications VI Problems & Perspectives, pages 741–749. Springer, 2011.

[23] P. Strachota and M. Beneš. Design and verification of the MPFA scheme for three-dimensional phase field model of dendritic crystal growth. In A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, and M. V. Tretyakov, editors, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, Leicester, September 2011, pages 459–467. Springer Berlin Heidelberg, 2013.

[24] P. Strachota, M. Beneš, and J. Tintěra. Towards clinical applicability of the diffusion-based DT-MRI visualization algorithm. J. Vis. Commun. Image R., 23(2):387–396, 2012.

[2] M. Beneš. Anisotropic phase-field model with focused latent-heat release. In FREE BOUNDARY PROBLEMS: Theory and Applications II, volume 14 of GAKUTO International Series in Mathematical Sciences and Applications, pages 18–30, 2000.

[3] M. Beneš. Mathematical and computational aspects of solidification of pure substances. Acta Math. Univ. Comenianae, 70(1):123–151, 2001.

[4] M. Beneš. Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. MathCzech., 48(6):437–453, 2003.

[5] M. Beneš. Computational studies of anisotropic diffuse interface model of microstructure formation in solidification. Acta Math. Univ. Comenianae, 76:39–59, 2007.

[6] J. C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, Chichester, 2003.

[7] J. Christiansen. Numerical solution of ordinary simultaneous differential equations of the 1st order using a method for automatic step change. Numer. Math., 14:317–324, 1970.

[8] L. Dagum and R. Menon. Openmp: An industry standard API for shared-memory programming. Computational Science & Engineering, IEEE, 5(1):46–55, 1998.

[9] R. Eymard, T. Gallouët, and R. Herbin. Finite volume methods. In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, volume 7, pages 715–1022. Elsevier, 2000.

[10] M. E. Gurtin. Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs. Oxford University Press, 1993.

[11] J. L. Gustafson. Fixed time, tiered memory, and superlinear speedup. In Proc. 5th Distributed Memory Computing Conference, pages 1255–1260, 1990.

[12] Intel Corporation. Intel true scale fabric architecture: Enhanced HPC architecture and performance, 2013. White paper.

[13] A. M. Meirmanov. The Stefan Problem. De Gruyter Expositions in Mathematics. Walter de Gruyter, 1992.

[14] Message Passing Interface Forum. MPI: A message-passing interface standard version 3.1, 2015.

[15] OpenMP Architecture Review Board. OpenMP application program interface version 4.0, July 2013.

[16] M. PunKay. Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In NNIN REU Research Accomplishments, pages 116–117. University of Michigan, 2005.

[17] S. L. R. E. Napolitano. Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system. Phys. Rev. B, 70(21):214103, 2004.

[18] W. E. Schiesser. The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego, 1991.

[19] A. Schmidt. Computation of three dimensional dendrites with finite elements. J. Comput. Phys., 125:293–3112, 1996.

[20] P. Stenström, T. Joe, and A. Gupta. Comparative performance evaluation of cache-coherent NUMA and COMA architectures. In ISCA ’92 Proceedings of the 19th annual international symposium on Computer architecture, volume 20, pages 80–91. ACM New York, NY, USA, May 1992.

[21] P. Strachota. Analysis and Application of Numerical Methods for Solving Nonlinear ReactionDiffusion Equations. PhD thesis, Czech Technical University in Prague, 2012.

[22] P. Strachota and M. Beneš. A multipoint flux approximation finite volume scheme for solving anisotropic reaction-diffusion systems in 3D. In J. Fořt, J. Fürst, J. Halama, R. Herbin, and F. Hubert, editors, Finite Volumes for Complex Applications VI Problems & Perspectives, pages 741–749. Springer, 2011.

[23] P. Strachota and M. Beneš. Design and verification of the MPFA scheme for three-dimensional phase field model of dendritic crystal growth. In A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, and M. V. Tretyakov, editors, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, Leicester, September 2011, pages 459–467. Springer Berlin Heidelberg, 2013.

[24] P. Strachota, M. Beneš, and J. Tintěra. Towards clinical applicability of the diffusion-based DT-MRI visualization algorithm. J. Vis. Commun. Image R., 23(2):387–396, 2012.